\section{Insurer Loss Distributions Vary With Portfolio Size} 
\label{sec:InsurerLossDistributionsVaryWithPortfolioSize} 

Two insurers, $\textbf{\textit{M}}$ and $\textbf{\textit{N}}$, randomly selecting $M$ and $N$ ($M >> N$) policyholders from the same population, with individual policyholder Population Loss Ratio Estimate standard deviation, $\sigma$, and mean loss ratio, $\mu$ = $PLR$ = 0.7500, draw their Population Loss Ratio Estimates from very different, normally distributed, Cumulative Population Loss Ratio Estimate Distribution Functions: $\Phi_{M}$($PLR$,$\frac{\sigma}{\sqrt{M}}$) and $\Phi_{M}$($PLR$,$\frac{\sigma}{\sqrt{N}}$), where $\frac{\sigma}{\sqrt{M}}$ $<<$ $\frac{\sigma}{\sqrt{N}}$ and each insurer's standard error determines all other insurers' standard errors since $\sigma_{e_{M}}$ = $\sigma$ * $\frac{\sqrt{N}}{\sqrt{M}}$. 

I will specify the Population Loss Ratio ($PLR$) and make reasonable, market appropriate, assumptions about Population Loss Ratio Estimate variation for a single, ``reasonably efficient'' Paradigm Insurer ($PI$). I then use these assumptions to analyze and compare the impact of portfolio size on operating results for four other insurers (See Table~\ref{tab:InsurerOperatingResultsByPortfolioSize}). I also provide more detailed tables in the appendices for a total of 35 different portfolio sizes. 

The ``Market Premium'' for this reasonably efficient insurer is an adequate, but not excessive, expense, risk, and profit loaded individual policyholder premium, such that if all policyholders pay the ``Market Premium'' to their insurers, the expected loss ratio for the entire population of potential policyholders will be equal to the population loss ratio, and the insurance market within which the Paradigm Insurer operates will be sustainable. Reasonable efficiency means that when the Paradigm Insurer receives the market premium, it can continue to operate with minimal risks of operating losses and insolvency, the Paradigm Insurer will be able to earn reasonable profits, and the policyholder benefits the Paradigm Insurer provides will be .

The Paradigm Insurer will earn profits, incur losses, and become insolvent with calculable probabilities based on the underlying variation in individual policyholders' standard deviations and the number of policyholders the Paradigm Insurer insures, through the Paradigm Insurer's standard error. 
